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The Stochastic Properties of High Daily Maximum Temperatures

Permanent Link: http://ufdc.ufl.edu/UFE0041390/00001

Material Information

Title: The Stochastic Properties of High Daily Maximum Temperatures
Physical Description: 1 online resource (61 p.)
Language: english
Creator: Keellings, David
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: amo, climate, crossing, florida, health, heat, model, predictive, stochastic, temperature
Geography -- Dissertations, Academic -- UF
Genre: Geography thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The statistical properties of the excursions of maximum daily temperatures above various critical thresholds of interest are analyzed with a view to developing models of heat wave events using more than 100 years of record from meteorological stations in Lake City, DeFuniak Springs, Avon Park, and Fort Myers, Florida. These stochastic variables include; event density (number of such events per unit time), duration, timing, and peak values over the threshold. The theoretical basis for the modeling is found in Crossing Theory, which states that as the threshold of interest becomes particularly large with respect to the mean of a Gaussian process, the number of crossings (up or down) becomes Poisson distributed. The changing seasonal intensity of such events can be incorporated by utilizing a non-homogeneous Poisson model, with time-varying rates. Environmental health studies indicate that both the magnitude and duration of the excursion above the critical threshold are important. As both the number of up-crossings and down-crossings follow a Poisson distribution, it is reasonable to approximate the length of time between the two (the duration of an event) by an exponential, or exponential-like distribution. Similarly, the peak magnitudes of event over the threshold (POT) represent the extreme tail of the distribution of daily maximum temperatures and might be assumed to follow the same sort of distribution. The current study only considers a single, arbitrarily defined, critical temperature threshold of 98degreeF (36.7degreeC). The threshold that constitutes a medically critical value may well vary spatially, and be dependent upon the ultimate application of the results. It is therefore necessary to be able to extrapolate findings derived at one level of interest to others, particularly in seeking to determine the risks of extremely rare events like those of Chicago (1995), France (2003), London and California (2006) and Melbourne (2009), which had seldom, if ever, been observed in historic series. The methodology has the flexibility to extrapolate to such levels while also having the advantage of being able to be applied to spatially differentiated data to determine risks associated with high temperature events during any time period or at any location of interest.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by David Keellings.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Waylen, Peter R.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041390:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041390/00001

Material Information

Title: The Stochastic Properties of High Daily Maximum Temperatures
Physical Description: 1 online resource (61 p.)
Language: english
Creator: Keellings, David
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: amo, climate, crossing, florida, health, heat, model, predictive, stochastic, temperature
Geography -- Dissertations, Academic -- UF
Genre: Geography thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The statistical properties of the excursions of maximum daily temperatures above various critical thresholds of interest are analyzed with a view to developing models of heat wave events using more than 100 years of record from meteorological stations in Lake City, DeFuniak Springs, Avon Park, and Fort Myers, Florida. These stochastic variables include; event density (number of such events per unit time), duration, timing, and peak values over the threshold. The theoretical basis for the modeling is found in Crossing Theory, which states that as the threshold of interest becomes particularly large with respect to the mean of a Gaussian process, the number of crossings (up or down) becomes Poisson distributed. The changing seasonal intensity of such events can be incorporated by utilizing a non-homogeneous Poisson model, with time-varying rates. Environmental health studies indicate that both the magnitude and duration of the excursion above the critical threshold are important. As both the number of up-crossings and down-crossings follow a Poisson distribution, it is reasonable to approximate the length of time between the two (the duration of an event) by an exponential, or exponential-like distribution. Similarly, the peak magnitudes of event over the threshold (POT) represent the extreme tail of the distribution of daily maximum temperatures and might be assumed to follow the same sort of distribution. The current study only considers a single, arbitrarily defined, critical temperature threshold of 98degreeF (36.7degreeC). The threshold that constitutes a medically critical value may well vary spatially, and be dependent upon the ultimate application of the results. It is therefore necessary to be able to extrapolate findings derived at one level of interest to others, particularly in seeking to determine the risks of extremely rare events like those of Chicago (1995), France (2003), London and California (2006) and Melbourne (2009), which had seldom, if ever, been observed in historic series. The methodology has the flexibility to extrapolate to such levels while also having the advantage of being able to be applied to spatially differentiated data to determine risks associated with high temperature events during any time period or at any location of interest.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by David Keellings.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Waylen, Peter R.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041390:00001


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1 THE STOCHASTIC PROPERTIES OF HIGH DAILY MAXIMUM TEMPERATURES By DAVID J. KEELLINGS A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010

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2 2010 David James Keellings

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3 To Cynthia and Jim, my pare nts, for their love and support

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4 ACKNOWLEDGMENTS I am very appreciative of Dr. Peter Waylen, Dr. Tim othy Fik, and Dr. Mark Brenner as members of my advisory committee and for their academic support in the completion of this research. I am especially grateful to Dr. Peter Waylen for his outstanding mentori ng and dedication to the growth of my academic career which has been invaluable throughout. I am also very grateful to Dr. Tim othy Fik who has afforded me much advice and encouragement. I am also indebted to the Geography Department for the support of its faculty and staff and to the University of Florida which has supported me financially with an Alumni Fellowship. I also want to express a special thanks to my very good friends and mentors Dr. Jeanne Fillman-Richards and Dr. Storm Richards who have supported me in a multitude of ways. They have been integral to my success and I will always regard them as family. Finally, I give my heartfelt thanks to my parents, Cynthia and Jim, for their infinite support and encouragement in the pursuit of my goals throughout my life and I also thank my Aunt Glenys for being there and providing accommodation to a wayward Scottish student.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...................................................................................................... 4 LIST OF TABLES ................................................................................................................ 7 LIST OF FIGURES .............................................................................................................. 8 ABSTRACT ........................................................................................................................ 10 CHAPTER 1 INTRODUCTION ........................................................................................................ 12 2 THE STOC HASTIC PROPERTIES OF HIGH DAILY MAXIMUM TEMPERATURES ...................................................................................................... 14 Introduction ................................................................................................................. 14 Study Locations and Data .......................................................................................... 15 Theory and Method ..................................................................................................... 16 Annual Event Density ........................................................................................... 17 Timing of Events within a Year ............................................................................ 17 First and Last Events of a Year ........................................................................... 18 Event Magnitude and Duration ............................................................................ 18 Critical Threshold ................................................................................................. 19 Independence of Events ...................................................................................... 19 Extrapolation to Higher Levels ............................................................................. 20 St atistical Stationarity and Climate Variability ........................................................... 20 Sea Surface Temperature Oscillations ............................................................... 20 Land/Water Contrast ............................................................................................ 22 Urban Heat Island ................................................................................................ 22 Results ........................................................................................................................ 23 Event Properties ................................................................................................... 23 Discrete Nature of Temperature Data ................................................................. 24 Extrapolation of Findings to Higher Levels ......................................................... 25 Discussion ................................................................................................................... 25 Normality of Temperature .................................................................................... 26 Division of the Record .......................................................................................... 26 Study Locations and Geographic Variability ....................................................... 27 Conclusions ................................................................................................................ 28 3 CONCLUSIONS .......................................................................................................... 43 APPENDIX: ALTERNATE LOCATION GRAPHS ......................................................... 45

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6 LIST OF REFERENCES ................................................................................................... 57 BIOGRAPHICAL SKETCH ................................................................................................ 61

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7 LIST OF TABLES Table page 2 -1 Observed parameters describing the relevant high temperature event characteristics. ........................................................................................................ 31

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8 LIST OF FIGURES Figure page 2 -1 Study station locations .......................................................................................... 31 2 -2 Computed probabilities of historic daily maxima exceeding a critical threshold of 95F (35 C) ....................................................................................................... 32 2 -3 Relative frequency of historical events compared to a Gaussian d istribution. ..... 33 2 -4 Historic mean daily maximum temperatures, one standard deviation. ............. 34 2 -5 Graphical definition of the stochastic variables (number, timing, magnitude and duration) and independence criteria ............................................................... 35 2 -6 Observed durations of heat wave events, in entire Lake City record, above a 98F ( 36.7C) threshold with f our day independence criteria. .............................. 35 2 -7 Time series of the annual number of observed heat wave events as defined by a 98F (36.7C) threshold and f our day independence criteria ....................... 36 2 -8 Observed and fitted Poisson distribution of ann ual numbers of heat wave events in DeFuniak Springs ................................................................................... 37 2 -9 Ob served POT magnitudes of events in DeFuniak Springs ................................. 38 2 -10 Observed durations of events in DeFuniak Springs ............................................. 39 2 -11 Probabilities of 0, 1, 2, 3, 4 events having occurred in DeFuniak Springs. .......... 40 2 -12 Probabilities of first events occurring, in DeFuniak Springs, before any day of the year and last events occurring after any day of the year. ............................... 40 2 -13 Comparison of observed and fitted event distributions. ........................................ 41 2 -14 Mean daily temperatures and precipitation from the historic al record. ................ 42 A-1 Observed and fitted Poisson distribution of ann ual numbers of heat wave events in Avon Park ............................................................................................... 45 A-2 Observed and fitted Poisson distribution of ann ual numbers of heat wave events in Fort Myers. .............................................................................................. 46 A-3 Observed and fitted Poisson distribution of ann ual numbers of heat wave events in Lake City. ................................................................................................ 47 A-4 Ob served POT magnitudes of events and fitted exponential distribution in Avon Park. .............................................................................................................. 48

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9 A-5 Observed POT magnitudes of events and fitted exponential distribution in Fort Myers. .............................................................................................................. 49 A-6 Ob served POT magnitudes of events and fitted exponential distribution in Lake City. ................................................................................................................ 50 A-7 Observed durations of events and fitted exponential distribution in Avon P ark ... 51 A-8 Observed durations of events and fitted exponential distribution in Fort Myers. ..................................................................................................................... 52 A-9 Observed dur ations of events and fitted exponential distribution in L ake City. ... 53 A-10 Probabilities of 0, 1, 2, 3, 4 events having occurred, in Avon Park, up to any day during the year. ............................................................................................... 54 A-11 Probabilities of 0, 1, 2, 3, 4 events having occurred, in Fort Myers, up to any day during the year. ............................................................................................... 54 A-12 Probabilities of 0, 1, 2, 3, 4 events having occurred, in Lake City, up to any day during the year. ............................................................................................... 55 A-13 Proba bilities of first events occurring, in Avon Park, before any day of the year and last events occurring after any day of the year. ..................................... 55 A-14 Probabilities of first events occurring, in Fort Myers, before any day of the year and last events occurring after any day of the year. ..................................... 56 A-15 Probabilities of first events occurring, in Lake City, before any day of the year and last events occurring after any day of the year. ..................................... 56

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10 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for th e Degree of Master of Science THE STOCHASTIC PROPERTIES OF HIGH DAILY MAXIMUM TEMPERATURES By David J. Keellings May 2010 Chair: Peter R. Waylen Major: Geography The statistical properties of the excursions of maximum daily temperatures above various critical thresholds of interest are analyzed with a view to developing models of he at wave events using more than 100 years of record from meteorological stations in Lake City, DeFuniak Springs, Avon Park, and Fort Myers, Florida. These stochastic variables include; event density (number of such events per unit time), duration, timing, and peak values over the threshold. The theoretical basis for the modeling is found in Crossing Theory which states that as the threshold of interest becomes particularly large with respect t o the mean of a Gaussian process, the number of crossings (up or down) becomes Poisson distributed. The changing seasonal intensity of such events can be incorp orated by utilizing a nonhomog eneous Poisson model, with timevarying rates. Environmental health studies indicate that both the magnitude and duration of the excursion above the critical threshold are important. As both the number of upcrossings and down -crossings follow a Poisson distribution, it is reasonable to approximate the length of time between the two (the duration of an event) by an exponential, or exponential like distribution. Similarly, the peak magnitudes of event

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11 over the threshold (POT) represent the extreme tail of the distribution of daily maximum temperatures and might be ass umed to follow the same sort of distribution. The current study only considers a single, arbitrarily defined, critical temperature threshold of 98 F (36.7C) T he threshold that constitutes a medically critical value may well vary spatially, and be dependent upon the ultimate application of the results. It is therefore necessary to be able to extrapolate findings derived at one level of interest to others, particularly in seeking to determine the risks of extremely rare events like those of Chicago (1995), France (2003), London and California (2006) and Melbourne (2009), which had seldom if ever been observed in historic series. The methodology has the flexibility to extrapolate to such levels while also having the advantage of being able to be ap pli ed to spatially differentiated data to determine risks associated with high temperature events during any time period or at any location of interest.

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12 CHAPTER 1 INTRODUCTION A scientific consensus now exists with regard to the acceptance of climate change but there is debate as to its cause and severity (IPCC, 2007). The global medical community has stated that climate change represents the most severe threat to human health across the globe dur ing the 21st century (Costello et al. 2009). Connections between climate and health are numerous and highly diverse. Many of these climate phenomena have obvious connections to health. They include severe storms, extreme temperatures, drought, and flooding, while some are less obvious and have consequences such as alterations in the ranges of infectious diseases and degradation of air quality. It is anticipated that climate change will cause extrem e events to occur with increased frequency and intensity (IPCC, 2007). To better plan for the adverse affects that climate change may have on human health, there is a need for information regarding how the climate will change both quantitatively and geogr aphically and how that change will be connected to health. Specifically, the global health community has outlined the need for accurate modeling with application to adaptable policy and response to health threats which will most likely have a varied soci etal context ( Costello et al. 2009). The e ffect that climate change will have on health will most likely vary geographically and as a function of population acclimatization and vulnerability. One such geographically varying impact on human health is the occurrence of high temperature events or heat waves. High temperatures cause overall death rates to increase, especially when the temperature rises above the local population's threshold or critical value (Curriero et al. 2002). In response to both the uncertainty of the magnitude of climate change and to

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13 the probable varied impacts that the change may have on humans this research focuses on the development of f lexible models that can quantify the risk of high temperature events and their associated properties. These models can be adapted to incorporate varied critical thresholds and are able to characterize the main stochastic variables that are associated with heat wave events. The stochastic variables investigated i nclude: event density (numbers of such events per unit time), duration, timing, and peak values over the threshold or the magnitude of events. In order to model these variables this study relies on Crossing Theory When a physical phenomenon, such as daily maximum temperatures, exhibits a distribution that is approximately Gaussian, many of its properties can be predicted using the simple statistical framework outlined in Crossing Theory (Rice, 1945; Leadbetter et al. 1983; Rodriguez -Iturbe and Bras, 19 85) In particular the theory is applicable to the estimation of physical event properties beyond a given level. Crossing Theory has previously been applied to studies of flood events above critical discharge thresholds as well as to the analysis of fr eeze probabilities (below threshold) with regard to agricultural impacts (Waylen, 1988 ; Waylen and LeBoutillier, 1989; Goto Maede et al. 2008). This research applies Crossing Theory to the development o f simple stochastic models that quantify the risk of high temperature events above a critical temperature threshold. The theory behind the models is explained and the models are applied to historic maximum daily temperature data spanning t he 20th century. The data come from fou r meteorological stations throughout Florida chosen to test the applicability of the models across varied geographic locations.

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14 CHAPTER 2 THE STOCHASTIC PROPERTIES OF HIGH DAILY MAXIMUM TEMPERATURES Introduction The occurrence of high temperature events or heat waves, poses a threat to human health, particularly to the more vulnerable segments of the population such as the very young and the elderly. Prior research suggests that extreme high temperatures are linked to increased levels of human morbidit y and mortality (Kunst et al. 1993; Hajat et al. 2002 ) and t he Centers for Disease Control and Prevention attributed an average of 688 deaths per year in the United States during 1999 -2003 to excessive heat (CDC, 2006) This number is greater than the tot al attributed to hurricanes tornadoes, floods, and winter storms combined over the same period. D eaths directly related to heat stress are, however, probably few in comparison to those tied to ot her medical conditions that are exacerbated by heat. The W orld Health Organization has identified preexisting health conditions or comorbidities, such as diabetes and obesity r espiratory or heart conditions which result in greater suscept ibility to heat wave events (World Health Organization Europe 1998). Moreover, the currently proposed global warming trend may result in warmer summer temperatures with increased frequencies, durations, and intensities of heat wave events (Gaffen and Ross 1998; Meehl and Tebaldi 2004). Numerous definitions of heat wave events exist in the literature, and the World Meteorological Organization (WMO) h as not yet defined the term (World Health Organization Europe 2009). The National Weather Service characterizes a heat wave as a period of abnormally and uncomfort ably hot and unusually humid weather with duration of at least two days (National Weather Service, 2009) Typically, if daily high

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15 temperature crosses a medically defined critical threshold for a specified number of days a heat wave is said to have occurred ( Tan et al. 2007 ). In order to be of epidemiological/medical significance any statistical approach to the study of heat waves must allow for fl exibility in the selection of an appropriate critical temperature and be capable of characterizing the du ration of events. Events occurring very early or late within the expected hot season may have greater epidemiological significance and therefore the timing of events is also an important consideration (Sheridan and Kalkstein, 2004). This study analyzed th e statistical properties of the excursions of maximum daily temperatures above various critical thresholds of interest with a view to developing flexible models of heat wave events These statistical properties or stochastic variables include: event densi ty (numbers of such events per unit time), duration, timing, and peak values over the threshold or the magnitude of events Study Locations and Data Four locations were selected for analysis in th is study based on the availability of long -term (> 100 years) and complete daily temperat ure records. The locations were also selected to include a broad geographic range throughout Florida (Figure 2-1). DeFuniak Springs is in the northwest, Lake City is in northern central, Avon Park is in southern cent ral, and Fort Myers is in the south. Fort Myers and DeFuniak Springs are both within 30 miles of the coast of the Gulf of Mexico while Avon Park and Lake City are both more than 60 miles from either the coast of the North Atlantic Ocean or the Gul f of Me xico. The locations also represent settings that have undergone extensive (Fort Myers) or relatively limited (Avon Park, DeFuniak Springs, Lake City) urban growth.

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16 H istoric datasets were obtained from t he Southeast Region Climate Data Center (SRCDC) The meteorological record for each station includes daily maximum, minimum, and mean temperature from the 1890s to 2008. All datasets were partially incomplete upon acquisition but all were deemed large enough to allow some years with missing data to be removed prior to the analysis. T o reduce data gaps a critical seasonal period with greater likelihood of high temperatures was identified. This season was defined by fitting a normal distribution to each days (1365) observed data from Lake City and c onsidering only those days returning a probability of 0.10 or greater that the maximum da ily temperature would exceed 95F (3 5 C) (Figure 2-2 ). This approach resulted in a critical high temperature band from June 1st through September 6th. Years missing data within this critical band were removed from subsequent analysis at all four locations. Theory and Method The approach used in this study to model high temperature events is based on Crossing Theory. Crossing Theory allows for the description of the statistical properties of events beyond some high temperature threshold and permits the probabilistic description of the number, timing, magnitude and duration of such events (Rice, 1945; Leadbetter et al. 1983; Rodriguez -Iturbe and Bras, 1985). Similar methodology has been applied to studies of flood events above a critical discharge threshold as well as to the analysis of freeze probabilities (below threshold) with regard to agricultural impacts (Waylen, 1988; Waylen and LeBoutillier, 1989; Goto Maede e t al. 2008). Crossing Theory states that as a threshold of interest becomes particularly large with respect to the mean of a Gaussian process, the number of crossings (up or down) becomes Poisson distributed. As both the number of up -crossings and dow n -crossings

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17 follow a Poisson distribution it is reasonable to approximate the length of time between the two (the duration of an event) by an exponential, or exponential -like distribution. Similarly, the peak magnitudes of event over the threshold (POT) represent the extreme tail of the distribution of daily maximum temperatures and might be assumed to follow the same sort of distribution. Basing the approach to modeling event properties on Crossing Theory provides this study with a strong theoretical basis while also allowing for the prediction of the statistical properties of events above progressively higher, and seldom encountered, t hresholds (Waylen and Woo, 1983; Birikundavyi and Rousselle, 1997). Annual Event Density The probability mass function of a Poisson distribution is given by: er of events per year: where K is the total number of events in N years of historical record. Timing of Events within a Year The Poisson distribution assumes that events are equally likely within a time period, in this case a year. However this assumption is unrealistic as heat wave events are strongly seasonal in their nature (Figure 2 -2). The Poisson distribution should therefore be modified to include a timedependent or nonhomogeneous function: / ) ( M e M PM N K / / ) ( ) ) ( () (n t e n t m Pn t

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18 and n is the number of events up t o that time in a year. T he modeling of distributions of the timings of events can be accomplished through estimat distribution (Figure 2-3) : where ) : ( t G is a Gaussian di stribution fitted to the timing of events with being mean date of exc eedance s and their standard deviation. First and Last Events of a Year The probability distributions of the dates of first, F(x (x of the year can be calculated from the time-dependent function shown above as demonstrated by Waylen (1988): Event Magnitude and Duration As POT events represent the extreme tails of daily maximum temperatures it is appropriate to represent the probability distribution of this event property with an exponential or exponential like function (Rousselle, 1972 ; Taesombut and Yevjevich, 1978): where X is the POT or magnitude of the event over the threshold and x is the mean POT or magnitude above the threshold. The duration of events represents the length of time between successive upward crossings of the daily maximum temperature above the threshold and downward ) : ( ) ( t G t )) ( exp( 1 ) ( t t x F )) ( ( exp( 1 ) ( t t xL ) / exp( 1 ) ( x X X x F

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19 crossings of the daily maximum temperature below the threshold. It is reasonable to assume that the duration of events will follow an exponential distribution (Cramer and Leadbetter 1967): where D is the duration of the event and d is the mean duration of all events above the threshold. Critical Threshold The critical threshold selected for analysis should be sufficiently far above the mean to satisfy the requirements of Crossing T heory while remaining at such a level as to maximize the number of heat wave events identif ied within the record. In this study the threshold is set at 98F (36.7C) which is at least one standard deviation above the daily mean maximum temperature for any day of the year (Figure 2 4). The choice of this threshold is somewhat arbitrary, other than to satisfy the statistical requirement referred to above, as the threshold which constitutes a threat to human health will vary geographically and will ultimately need to be medically defined. Independence of Events In the present study events are c onsidered to be independent when the minimum number of days separating two consecutive events exceeds four days. Events separated by four days or less of sub -critical threshold temperatures are gr ouped ( Figure 2 5 ). An i ndependence criterion was set in this way to account for the possible epidemiological significance of sub-critical threshold relief days between events. M edical literature confirm s this choice because only a weak association between heat -related mortality on any given day and temperatures more than four days ) / exp( 1 ) ( d D D dF

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20 prior has been shown (Curriero et al. 2002). The use of independence criteria creates many longer -duration or compounded events within the series. T he majority of the longer -duration events included in the analysis are the result of compounding multiple shorter duration events (Figure 2-6) Extrapolation to Higher Levels It is possible to extrapolate from the distributions employed in this study to determine the probabilities associated with event p arameters above higher truncation levels or critical thresholds. As exponential distributions are "memoryless any portion of the distribution is itself exponentially distributed and described by the same parameter (Ross, 1976). Therefore, as the critic al threshold is raised the mean parameter remains constant. The expected number of events with a certain magnitude can be approximated using the estimated exponential functi on of magnitude outlined above. Subsequently, if the exponential distribution is a good representation of the magnitudes of events above a particular critical threshold, and therefore the proportion of events that will survive changes in the critical level, then the number of crossings above any critical threshold can also be estimated. The seasonal characteristic of events (timing within a year, probabilities of first an d last events) can be estimated as outlined above, using the estimated number of crossings in combination with the lower -threshold-generated Gaussian distribution as its mean and variance remain constant as the truncation level is increased. Statistical Stationarity and Climate Variability Sea Surface Temperature Oscillations The statistical approach used in this study assumes that the values of the model parameters remain invariable or stationary through out the period of study However,

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21 variability arising from macro -scale fluctuations in climate operating at low frequencies may partially control event risk. Such low -frequency variability may be associated wit h sea surface temperature (SST) oscillations. The El Nio-Southern Oscillation (ENSO) and the Atlantic Multi decadal Oscillation (AMO) appear to have a significant influence on Floridas climate ( GotoMaede et al. 2008). There is also recent research su ggesting that significant Atlantic SST phases exist at the decadal time scale on a global as well as regional scale (Dong and Sutton, 2002; Sutton and Hodson, 2005; Zhang and Delworth, 2005) and that they can be linked to a separate phenomenon known as the North Atlantic Oscillation (NAO) (Wu and Gordon, 2002; Justino and Peltier, 2005) Research suggests clear links between ENSO phase and winter temperatures and precipitation in Florida (Gershunov and Barnett, 1997) but ENSO phase appears to have littl e effect on summertime high temperatures and therefore heat wave occurrence Studies of the impact of the AMO on precipitation over eastern North America abound in the literature (Enfield et al 2001, Knight et al 2006, McCabe and Palecki, 2006) They have shown that precipitation is significantly reduced during warm phases of the AMO but that the effect is variable in both magnitude and spatial extent. The AMO appears to have a significant influence on mean air temperatures in both North America and Western Europe ( Kerr, 2000 ; Sutton and Hodson, 2005; Goto Maede et al. 2008 ; Arguez et al. 2009 ). The high or warm phase (above average) of Atlantic SSTs has been associated with mean temperatu re anomaly increases of up to 1.5C ( ~ 3 F) in the North Atlantic region with peaks in these anomalies occurring during the summer months of June, July, August, and September

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22 (Arguez et al. 2009). Enfield calculated an index of the AMO from detrended long run averages of mean SST observations from 1860 to 1994 and identified periods in which the AMO has been in a warm phase or cool phase relative to the mean. Land/Water Contrast Another major factor in the control and distribution of temperature across Florida is land/water contrasts. Florida is a peninsula with the Atlantic Ocean to the east and the Gulf of Mexico to the west, and land near a large body of water is influenced by the temperature of the water. Water has a high specific heat capacity and allows energy to penetrate to a greater dept h than on land. It has the ability to mix both vertically and horizontally as well as a greater capability for evaporative cooling. Therefore, land heats and cools more rapidly than water. In summer months, coastal areas tend to be cooled when air from above the water moves over the land, displacing warmer air and resulting in what is referred to as a sea breeze. Despite Floridas proximity to large bodies of water, cool sea breezes do not entirely regulate temperatures across the peninsula. Howev er, a temperature gradient does exist as one moves inland. For example, the following are July average temperatures observed at meteorological stations in south Florida: Miami Beach 30.8 C (87.4 F ), Miami International Airport (8 miles inland) 31.9 C (89.5 F ), Bend-Tamiami Trail (40 miles in land) 33.3 C (92.0 F ) (Winsberg & Simmons, 2009). Urban Heat Island The urban heat island (UHI) phenomenon has also been proposed to operate as a climate control that a ffects local temperature variation. Floridas population has grown tremendously over the last several decades and this growth has resulted in urbanization and land cover change. Much of the developed coastal areas of Florida and the central

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23 I -4 corridor have merged together to form what is referred to as a megalopolis of connected urban areas. U rban areas are typically warmer than more open naturally vegetated areas due to the presence of more energy absorptive and radiative manm ade surfaces. A recent 36month study of Orlando found that mean d aily maximum temperatures of areas in the city center and those on the outskirts could vary by as much as 8 C (14 F ) (Yow & Carbone, 2006). Results Event Properties After investigation o f likely non -stationarity in the record, as outlined above, the record was consid ered to be composed of different statistical populations. Empirical evidence suggests that a primary determinant of the climate variability in Florida may arise from fluctuations in the AMO. The annual time series of the number of heat wave events at the four study locations (Figure 2 -7) indicates a high degree of inter annual variability and tendency for a low -frequency clustering of similar total numbers of events with some correspondence to AMO ph ase as defined by Enfield et al. (2001 ). Therefore, to accommodate this physically driven and empirically observed variability the data were divided by AMO pha se for subsequent analysis. Application of the Poisson distribution to the modeling of annual event density is satisfactory (Figure 2-8) as is the a pplication of the exponential distribution to the magnitude and duration of satisfactory events (Figure 2-9; Figure 210) Using a one -sample Kolmogorov -Smirnov goodness of -fit test (Crutcher 1975) the null hypothesis of no significant difference between observed and predicted frequencies cannot be rejected. The test was carried out at the 0.80 significance level to minimize the risk of type II errors. From these results it is possible to predict the probabilities of: ( 1) the annual density of events,

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24 (2) the magnitu des of events above a threshold, and (3) events exceeding a given duration. Utilization of the nonhomogeneous Poisson function allows for prediction of the probability of observing 0,1,2,3 and up to m events on an y given day of the year (Figure 2 11 ). The combination of the exponential distribution and the time-dependent function ( ) ( t ) is applied to calculate probabilities of first and last events of the year (Fi gure 2 12 ). Discrete Nature of Temperature Data It is worth exploring the discrete nature of the temperature data. In reality temperature is a continuous variable, but the data were rendered discrete (e.g. 98 F 99 F 100 F ). This may be the result of the use of crude instrumentation or rounding but the true ranges these discrete temperature values represent cannot be stated with certainty We could assume, for example, that any temperature greater than 98 F but less than or equal to 99 F (e.g. 98.01, or 99.00) was recorded as 99 F If we assume that the group of temperatures within this range (98.0199.00) is uniformly distributed, we may represent them as being 0.5 F (the mean of the group) above the 98 F threshold. Therefo re, the dis crete value is shifted to 98.5 F from 98 F to better represent a continuous range of temperature. Such a shift in temperature values would clearly impact parameters in the analysis and the relationship to fitted functions. The observed data and mean whi ch is used to compute fitted functions would be shifted 0.5 F lower. However, we may also assume that any temperature greater than 98.5 F but less than or equal to 99.5 F was recorded as 99 F and that, if the prior assumptions are correct, the mean is also 99 F and therefore the discrete value is representative of a continuous range. Given the uncertainty as to the nature of the recorded temperature

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25 values we are therefore unable to relate them to specific real continuous ranges and cannot justify a r evaluation of the data away from the discrete record. Extrapolation of Findings to Higher Levels To test the ability of the methodology to extrapolate findings from one critical threshold to a higher or seldom encountered threshold data from 19661994 in DeFuniak Springs are used as an example. If the critical threshold is raised to 99 F (3 7.2C) we would expect 64.4% (Figure 2-9) of the original 22 events to survive the increase in the threshold. The expected number of events remaining would therefore be 14.17 and the observed number is 11. We would also expect the mean magnitude of events to remain constant at 2.27 as the threshold is increased. The observed mean is 2.55. Raising the critical threshold to 100F (3 7.8C) we would expect 41.5% (9.13 events) of the original 22 events to survive and the observed number is 7, while the mean is 2.43 Pushing the threshold to seldom observed levels (102F / 3 8.9C ) we would expect only 17.2% (3.78 events) of the original event s to remain and 3 are observed while the mean is 2. Expected number of events can then simply be divided by the 21 years of record to yield the mean number of events per year or The application of a Kolmogorov Smirnov test confirms that this is a stat istically acceptable approach at the 0.80 significance level. Discussion The approach to modeling high temperature or heat wave events illustrated in this study is capable of statistically describing the stochastic variables associated with such events an d also allows for a flexible definition of the criteria which constitute event occurrence. The probability distributions employed are well known and the methods used have strong theoretical support. Other more complex exponential -like distributions

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26 (gene ralized Pareto, Gamma) may be used as alternatives to the simple exponential distribution employed in this study. However, such alternatives were investigated and found to add minimal or no improvements to the models so in the interest of parsimony they w ere not included in this study. Normality of Temperature The methods employed rely on the reasonable assumption of normality of temperature distribution on any given day. The distribution of the timings of events has also been shown to be statistically s imilar to that of a normal distribution (Figure 2 -3). T here are however some notable deviations from normality particularly in the months of June and July (F igure 2 13 ). A possible explanation for the noted deviation may lie in seasonal climate variabili ty associated with precipitation. M ean daily precipitation from the entire record in Lake City, FL may have some impact upon the timing of the temperature deviations from simple normality (Figure 2-14) The drier period through the m onths of April and M ay and the consequent lack of atmospheric moisture and cloud cover (albedo) may contribute to greater insolation and higher temperatures. Through the month of June and int o July precipitation increases greatly as the wet season begins. Increased precipi tation, resulting in greater atmospheric moisture and cloud cover may lead to decreased insolation and relatively cooler maximum daily tem peratures. In this respect, fluctuations in precipitation during these months can be seen to correspond to the deviation of events from the fitted normal distribution. Division of the Record The data were divided based on phase of the AMO and much of the modeling was conducted using parameters calculated from these sub-samples. This approach allowed for the considerat ion of the existence of non-stationarity and subsequently

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27 differentiated statistical sub populations within the historical record. The division of the record in this manner serves two purposes. Firstly, it is necessary due to abundant physical evidence of climate variability which, according to prior research, appears to be at least part ial ly controlled by the phase of the AMO. G raphical results along with the distribu tion parameters illustrate the periodic variability of the high temperature event properties during alternate phases of the AMO (Table 2 1) The mean parameters of event density, POT, and duration are all generally elevated during the 19261965 period which corresponds to the warm phase of the AMO. Although the current study is not focused on the investigation of the relationship between high temperature event parameters and the phase of the AMO it is interesting that the noted pattern appeared and is in line with previous research into the AMO impa ct on mean air temperature Secondly, the division of the record by phase of the AMO is a convenient test of the flexibility of the study method itself. The application of the method to these varied climate periods illustrates the suitability of the method to the an alysis of varied climates and a s a flexible tool for predicting future high temperature or heat wave event properties under the influence of projected climate change scenarios. Study Locations and Geographic Variability For this study geographica lly dist inct locations were selected and analyzed to encompass the variability outlined above. As the goal of this study is to develop flexible models of the stochastic properties associated with high temperature events that can be adapted to any l ocation, it doe s not focus on relating high temperature events to any particular meteorological, climatological or anthropogenic phenomenon. The main contribution of this work lies in the presentation of a robust statistical methodology that can be applied under varied conditions to render probabilistic estimates of the

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28 stochastic variables associated with high temperature events. Nevertheless it is worth discussing the influence of geographic variability on the occurrence of high temperatures observed at the four stud y locations around Florida, because the model parameters reflect a great deal of information about changes in physical environments Latitude has less of an effect on high temperature occurrence than one might expect. For exampl e, Fort Myers has a relati vely lower frequency of events and those with a high POT are rare compared to other locations despite being the most southerly, and the most urbanized, of the fo ur Fort Myers' proximity to the ocean appears to be the dominant control on the occurrence o f high temperatures at this location. The more northern locations of DeFuniak Springs and Lake City exhibit greater frequencies of high temperature events which are typically of higher magn itude and longer duration (Table 2 1 ). Conclusions Heat wave events are caused by several physical climate processes that interact to produce dangerous conditions for human health. The interaction of these factors makes the study of hea t waves complex and has led to the use of heat indices. H igh temperature however remains the major factor for defining a heat wave event. High humidity place s additional stress on human populations during high temperature events but it was not considered in this study as there is a lack of historic humidity data. During the summer or "hot season," Florida is typically under the influence of prevailing southerly wind flow. Winds coming from the south over the peninsula carry warm and typically moist tropical air. Therefore, during the high temperature season in Florida, there is g enerally constant high humidity so this factor c an be considered a controlled variable. High temperatures in Florida are influenced by the s tates southern location

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29 and proximity to the ocean but SST oscillations may exert control over much of the variability in high temperatures within the state. The ever -increa sing urbanization and associated land cover change throughout Florida have been shown to increase local maximum temperatures (Marshall et al. 2004) Additionally, with climate change scenarios predicting increasing temperatures the future may bring greater increased heat wave probabilities ( Meehl and Tebaldi 2004). T here will be great spatial variability in the response of regional climate to probable climate change scenarios as suggested by an ensemble of model outputs (Kerr, 2008). The eastern U.S. is not expected to be highly responsive to probable climate change, but southern Florida is a not able exception and is predicted to be relatively responsive to climate change (Kerr, 2008). The current study sought to develop flexible models that incorporate the stochastic properties associated with the physical phenomenon of high temperature events. The occurrence of such events places stress on the human population and therefore involves potential medical consequences. These events also stress vegetation as well as the often overlooked animal community The temperature t hreshold that constitutes a threat to human health will certainly vary spatially and be influenced by physical, social, and even behavioral factors including the built environment, access to cooling technology, level of acclimatization to heat, and physical activity. These factors are currently the focus of much research in the climate and health community and critical thresh olds will ultimately need to be medically defined. The methodology utilized in this study is based on deviation from the mean and although the critical threshold may vary in many respects the statistical relationship

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30 used to model events has the flexibili ty to extrapolate to differing medically defined levels. Using Crossing Theory to model event properties provided this study with a strong theoretical basis while also allowing for the prediction of the statistical properties of events above progr essively higher, and seldom enc ountered, thresholds (Waylen & Woo, 1983, Birikundavyi & Rousselle, 1997). The ability to extrapolate findings derived at one level of interest to others is of particular importance when seeking to determine the risks of ext remely rare events like those of Chicago (1995), France (2003), London and California (2006) and Melbourne (2009), which had seldom if ever been observed in historic climate data. The methodology has the flexibility to extrapolate to such levels while al so having the advantage of being applicable to spatially differentiated data to determine risks associated with high temperature events during any time period or at any location of interest.

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31 Table 2 1. Observed parameters describing the relevant high temperature event characteristics obtained from the four study locations subdivided by AMO cool/warm phase periods as defined by Enfield et al. ( 2001 ). Meteorological Station Time period Events/Year Mean POT Mean Duration Avon Park (Cool AMO) (Warm AMO) (Cool AMO) DeFuniak Springs Fort Myers Lake City 1905 1925 19261965 19661994 19051925 19261965 19661994 19051925 19261965 19661994 19051925 19261965 1966 1994 0.11 0.81 0.72 1.57 2.21 1.05 0.00 0.13 0.14 1.11 1.37 0.81 1.00 2.27 1.33 2.45 2.58 2.27 0.00 2.20 2.00 1.95 2.37 2.14 1.00 2.50 2.50 1.64 4.04 3.32 0.00 4.00 3.75 2.53 3.15 2.19 Figure 2 1. State of Florida showing locations of meteorological stations used in the study. x d

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32 Figure 2 2 Computed probabilities of historic daily maxima exceeding a critical threshold of 95 F (3 5 C) Dashed lines represent critical high temperature band defined based on Lake City data. 0 0.05 0.1 0.15 0.2 0.25 0.3P(Tmax>95 F)Avon Park Computed Daily Probability 10 Day Moving Average 0 0.05 0.1 0.15 0.2 0.25 0.3P(Tmax>95 F)DeFuniak Springs 0 0.05 0.1 0.15 0.2 0.25 0.3 P(Tmax>95 F)Fort Myers 0 0.05 0.1 0.15 0.2 0.25 0.3 J F M A M J J A S O N D P(Tmax>95 F)MonthLake City Critical Band

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33 Figure 2 3. Relative frequency of historical events compared to a Gaussian distribution. Observed and predicted values do not differ at 0.80 significance level. Events are defined as exceeding a 98 F (36.7C) threshold and events have been grouped if separated by a maximum of four days of sub-critical temperatures. 0 0.01 0.02 0.03 0.04 0.05 0.06Relative FrequencyAvon Park 0 0.01 0.02 0.03 0.04 0.05 0.06Relative FrequencyDeFuniak Springs 0 0.01 0.02 0.03 0.04 0.05 0.06 Relative FrequencyFort Myers 0 0.01 0.02 0.03 0.04 0.05 0.06 60 90 120 150 180 210 240 270 300 Relative Frequency Day of YearLake City

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34 Figure 2 4 Historic mean daily maximum temperatures, one standard deviation. 50 60 70 80 90 100Temp ( F)Avon Park 50 60 70 80 90 100Temp ( F)DeFuniak Springs 50 60 70 80 90 100Temp ( F)Fort Myers 50 60 70 80 90 100 1 31 61 91 121 151 181 211 241 271 301 331 361 Temp ( F)Day of YearLake City

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35 Figure 2 5 Graph ical definition of the stochastic variables (number, timing, magnitude and duration) and independence criteria employed to describe heat waves cros sing a prescribed critical threshold. Figure 2 6. Observed durations of heat wave events in entire Lake City record, above a 98F (36.7C) threshold with four day independence criteria. Events are plotted by date of start. Red points are compounded based on the independence criteria. 0 2 4 6 8 10 12 14 90 120 150 180 210 240 270 300Duration (days)Day of Year Compounded Events Day of Year Maximum Daily Temperature ( o F) 84 86 88 90 92 94 96 98 100 102 m(t) = 0 m(t) = 1 m(t) = 2 m(t) = 3d 1 =3 d 2 =8 d 3 =1 x 1 =4 x 2 =2 x 3 =1 a 1 =183 a 2 =192 a 3 =207m(t) = Events up to day t a = Starting dates of events (day of year) d = Duration of Events (days) x = Peak magnitude of event above threshold ( o F) T crit = 97 o FJul. 1 Jul. 11 Jul. 21 Jul. 31 Aug. 10 Jun. 24

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36 Figure 2 7. Time series of the annual number of observed heat wave events as defined by a 98 F (36.7C) threshold and four day independence criteria. Value of -1" denotes missing data. Colored arrows represent the phase of the AMO as defined by Enfield et al. (2001 ). 1 1 3 5 7 9 1895 1902 1909 1916 1923 1930 1937 1944 1951 1958 1965 1972 1979 1986 1993 2000 2007 Number of EventsAvon Park 1 1 3 5 7 9 1897 1904 1911 1918 1925 1932 1939 1946 1953 1960 1967 1974 1981 1988 1995 2002 Number of EventsDeFuniak Springs 1 1 3 5 7 9 1892 1902 1912 1922 1932 1942 1952 1962 1972 1982 1992 2002 Number of EventsFort Myers 1 1 3 5 7 9 1893 1903 1913 1923 1933 1943 1953 1963 1973 1983 1993 2003Number of EventsYearLake City Coo l War m Coo l Warm

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37 Figure 2 8. Observed and fitted Poisson distribution of annual numbers of heat wave events, in DeFuniak Springs, above a 98 F (36.7C) threshold with four day independence criteria. For the corresponding graphs of alternate locations please see Appendix A. 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 7 8 Probability19051925 (COOL AMO) Observed Poisson 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 7 8 Probability19261965 (WARM AMO) 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 7 8 ProbabilityHeat Events/Year 19661994 (COOL AMO) = 1.57 = 2.21 = 1.05

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38 Figure 2 9. Observed POT magnitudes of events, in DeFuniak Springs, above 98 F (36.7C) threshold with four day independence cri teria and fitted exponential distribution. For the corresponding graphs of alternate locations please see Appendix A. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10Exceedance Probability19051925 (COOL AMO) Observed EXP 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10Exceedance Probability19261965 (WARM AMO) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10Exceedance ProbabilityPOT ( F) 19661994 (COOL AMO) 45 2 x 58 2 x 27 2 x

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39 Figure 2 10. Observed durations of events, in DeFuniak Springs, above 98 F (36.7C) threshold with four day independence criteria and fitted e xponential distribution. For the corresponding graphs of alternate locations please see Appendix A. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22Exceedance Probability19051925 (COOL AMO) Observed EXP 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22Exceedance Probability19261965 (WARM AMO) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22Exceedance ProbabilityDuration (Days) 19661994 (COOL AMO) 64 1 d 04 4 d 32 3 d

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40 Figure 2 11. Probabilities of 0, 1, 2, 3, 4 events having occurred, in DeFuniak Springs, up to any day during the year. For the corresponding graphs of alternate locations please see Appendix A. Figure 2 12. Probabilities of first events occurring, in DeFuniak Springs, before any day of the year and last events occurring after any day of the year. For the corresponding graphs of alternate locations please see Appendix A. 0.0 0.2 0.4 0.6 0.8 1.0 100 120 140 160 180 200 220 240 260 280 P(m(t) = n)Day of Year (t) 0 0.2 0.4 0.6 0.8 1 120 140 160 180 200 220 240 260 P(before & after x)Day of Year First Event Last Event m(t) = 0 m(t) = 1 m(t) = 2 m(t) = 3 m(t) = 4

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41 Figure 2 13. Comparison of observed and fitted event distributions. A) Observed cumulative probability of events exceeding 98 F (36.7C) threshold with four day independence criteria and fitted cumulative normal distribution. B) Deviation of observed event probability from normal distribution. Apr May Jun Aug Jul Day of Year 90 120 150 180 210 240 270 Cumulative Probability 20 40 60 80 Fitted Observed Sep 90 120 150 180 210 240 270 Deviation from Normal Dist. -0.020 -0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020 Deviation 29 day running mean Apr May Jun Aug Jul Sep A B

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42 Figure 2 14. Mean daily temperatures and precipitation from the historical record. Day of Year 0 30 60 90 120 150 180 210 240 270 300 330 360 Daily Temperature (F) 40 50 60 70 80 90 Mean Daily Precipittaion (ins) 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Mean Daily Max T Mean Daily Min T Mean Daily P 29 day running mean

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43 CHAPTER 3 C ONCLUSIONS Simple stochastic models were presented to characterize the major properties associated with high temperature events above a critical threshold. These models were developed using strong theoretical support found in Crossing Theory. C omparison of the models with historic data indicates that they satisfactorily estimate risks of each of the high temperature event properties investigated. The application of these models to varied geographic locations around F lorida supports the notion that the models are capable of determining the risks associated with high temperature events through time and across spatially and climatologically distinct locations. The focus of this research was on modeling high daily maxim um temperatures above a critical threshold. Other meteorological variables such as high humidity and poor air quality may increase the stress placed on human populations during high temperature events. Previous research examined the use of heat indices which attempt to quantify human discomfort (Kalkstein and Valimont, 1986). Public health warning systems or Heat Health Warning Systems (HHWS) have been developed using air mass synoptic classification e.g the Philadelphia Hot Weather -Health Watch/Warn ing System (PWWS) (Sheridan and Kalkstein, 1998). These systems use meteorological forecasts to predict the likelihood of combinations of conditions (air temperature, dew point temperature, cloud cover, sea level pressure, wind speed, and wind direction) that are expected to have adverse impacts on health. It has been found that air masses with particular combinations of these conditions are significantly correlated with increased mortality. O f all the variables investigated, however, maximum temperature s and duration of air mass presence were among the most highly

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44 correlated with increased mortality ( Kalkstein et al. 1996). The significance of temperature has also come to light in another study which found daily average temperature to be a stronger pr edictor of mortality than dew point (Curriero et al. 2002). The development of flexible stochastic models in this research is intended as the first step in an on-going study that seeks to establish associations between heat wave event properties and human mortality. These associations can then be combined with heat wave event characteristics derived from future climate change scenarios to give predictions of heat wave events and their associated mortality. The development of an online heat risk mapping tool is envisioned that will combine all three of these elements (statistical theory outlined in this research, mortality associations, climate change scenarios) with web -based programming and Geographical Information Systems. Such a tool would allow epidemiological/medical users to specify their own heat wave criteria and generate risk maps across an area of interest. Through combination with other geographical databases it may also be possible to include other elem ents such as demographic, socio economic, and lifestyle data with climate and mortality relations to indicate areas that may be at higher risk. The future development of a heat risk mapping tool will rely heavily on th e stochastic models outlined in this research. Unlike the existing HHWS which are limited to five day forecasts, this modeling provides lo ng term predictive ability that will allow a proactive rather than reactive approach to future policy decisions.

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45 APPENDIX ALTERNATE LOCATION G RAPHS Figure A 1 Observed and fitted Poisson distribution of annual numbers of heat wave events, in Avon Park, above a 98F (36.7C) threshold with four day independence criteria. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 Probability19051925 (COOL AMO) Observed Poisson 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 Probability19261965 (WARM AMO) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 ProbabilityHeat Events/Year 19661994 (COOL AMO) = 0.11 = 0.81 = 0.72

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46 Figure A 2 Observed and fitted Poisson distribution of annual numbers of heat wave events, in Fort Myers, above a 98F (36.7C) threshold with four day independence criteria. Note that during the 1905-1925 (Cool AMO) period there were no observed events above a 98F (36.7C) threshold. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 Probability19261965 (WARM AMO) Observed Poisson 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 ProbabilityHeat Events/Year 19661994 (COOL AMO) = 0.13 = 0.14

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47 Figure A 3 Observed and fitted Poisson distribution of annual numbers of heat wave events, in Lake City, above a 98F (36.7C) threshold with four day independence criteria. 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 7 8 9 Probability19051925 (COOL AMO) Observed Poisson 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 7 8 9 Probability19261965 (WARM AMO) 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 7 8 9 ProbabilityHeat Events/Year 19661994 (COOL AMO) = 1.11 = 1.37 = 0.81

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48 Figure A 4 Observed POT magnitudes of events in Avon Park above 98F (36.7C) threshold with four day independence criteria and fitted exponential distribution. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6Exceedance Probability19051925 (COOL AMO) Observed EXP 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6Exceedance Probability19261965 (WARM AMO) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6Exceedance ProbabilityPOT ( F) 19661994 (COOL AMO) 00 1 x 27 2 x 33 1 x

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49 Figure A 5 Observed POT magnitudes of events in Fort Myers above 98F (36.7C) threshold with four day independence criteria and fitted exponential distribution. Note that during the 19051925 (Cool AMO) period there were no observed events above a 98F (36.7C) threshold. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6Exceedance Probability19261965 (WARM AMO) Observed EXP 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6Exceedance ProbabilityPOT ( F) 19661994 (COOL AMO) 20 2 x 00 2 x

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50 Figure A 6 Observed POT magnitudes of events in Lake City above 98F (36.7C) threshold with four day independence criteria and fitted exponential distribution. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9Exceedance Probability19051925 (COOL AMO) Observed EXP 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9Exceedance Probability19261965 (WARM AMO) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9Exceedance ProbabilityPOT ( F) 19661994 (COOL AMO) 95 1 x 37 2 x 14 2 x

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51 Figure A 7 Observed durations of events in Avon P ark above 98F (36.7C) threshold with four day in dependence criteria and fitted exponential distr ibution. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13Exceedance Probability19051925 (COOL AMO) Observed EXP 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13Exceedance Probability19261965 (WARM AMO) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13Exceedance ProbabilityDuration (Days) 19661994 (COOL AMO) 00 1 d 5 2 d 5 2 d

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52 Figure A 8 Observed durations of events in Fort Myers above 98F (36.7C) threshold with four day independence criteria and fitted exponential distribution. Note that during the 19051925 (Cool AMO) period t here were no observed events above a 98F (36.7C) threshold. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Exceedance Probability19261965 (WARM AMO) Observed EXP 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Exceedance ProbabilityDuration (Days) 19661994 (COOL AMO) 00 4 d 75 3 d

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53 Figure A 9 Observed durations of events in L ake City above 98F (36.7C) threshold with four day in dependence criteria and fitted exponential distr ibution. 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13Exceedance Probability19051925 (COOL AMO) Observed EXP 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13Exceedance Probability19261965 (WARM AMO) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13Exceedance ProbabilityDuration (Days) 19661994 (COOL AMO) 53 2 d 15 3 d 19 2 d

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54 Figure A 10. Probabilities of 0, 1, 2, 3, 4 events having occurred, in Avon Park up to any day during the year. Figure A 11. Probabilities of 0, 1, 2, 3, 4 events having occurred, in Fort Myers up to any day during the year. 0.0 0.2 0.4 0.6 0.8 1.0 100 120 140 160 180 200 220 240 260 280P(m(t) = n)Day of Year (t) 0.0 0.2 0.4 0.6 0.8 1.0 100 120 140 160 180 200 220 240 260 280 P(m(t) = n) Day of Year (t) m(t) = 0 m(t) = 1 m(t) = 2 m(t) = 3 m(t) = 4 m(t) = 0 m(t) = 1 m(t) = 3 m(t) = 2 m(t) = 4

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55 Figure A 12. Probabilities of 0, 1, 2, 3, 4 events having occurred, in Lake City up to any day during the year. Figure A 13. Probabilities of first events occurring, in Avon Park, before any day of the year and last events occurring after any day of the year. 0.0 0.2 0.4 0.6 0.8 1.0 100 120 140 160 180 200 220 240 260 280 P(m(t) = n)Day of Year (t) 0 0.2 0.4 0.6 0.8 1 120 140 160 180 200 220 240 260 P(before & after x)Day of Year First Event Last Event m(t) = 0 m(t) = 1 m(t) = 2 m(t) = 3 m(t) = 4

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56 Figure A 14. Probabilities of first events occurring, in Fort Myers, before any day of the year and last events occurring after any day of the year. Figure A 15. Probabilities of first events occurring, in Lake City, before any day of the year and last events occurring after any day of the year. 0 0.2 0.4 0.6 0.8 1 120 140 160 180 200 220 240 260 P(before & after x)Day of Year First Event Last Event 0 0.2 0.4 0.6 0.8 1 120 140 160 180 200 220 240 260 P(before & after x)Day of Year First Event Last Event

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57 LIST OF REFERENCES Arguez A OBrien JJ, Smith SR. 2009. Air temperature impacts over Eastern North America and Europe associated with low -frequency North Atlantic SST variability. International Journal of Climatology 29: 1 10. Birikund avyi S, Rousselle J. 1997. Use of partial duration series for single stat ion regional analysis of floods. Journal of Hydrologic Engineering 2 : 68 -75. CDC. 2006. Heat -related deaths --United States, 19992003. MMWR Morbidity and Mortality Weekly Report 55(29): 796 -798. Costello A, Abbas M, Allen A, Ball S, Bell S, Bellamy R., et al. 2009 Managing the health effect s of climate change: Lancet and University College London Institut e for Global Health Commission. Lancet 373: 1693 -733. Cramer H, Leadbetter MR. 1967. Stationary and related stochastic processes Wiley: New York. Crutcher HL. 1975. A note on the possible misuse of the Kolmogorov -Smirnov test, Journal of Applied Meteorology 14 : 1600 -1603. Curriero FC Heiner KS Samet JM, Zeger SL, Strug L, Patz JA. 2002 Temperature and mortality in 11 cities of the eastern United States. American Journal of Epidemiology 155: 80 -87. Dong BW, Sutton RT. 2002 Adjustment of the coupled ocean atmosphere system to a sudden change in the thermohaline circulation. Geophysical Research Letters 29: 1728. Enfield DB, Mestas -Nunez AM, Trimble PJ. 2001. The Atlantic multidecadal oscillation and its relation to rainfall and river flows in the continental U.S. Geophysical Research Letters 28: 2077 -2080. Gaffen DJ Ross R J. 1998. Increased summertime heat stress in the US. Nature 396: 529-530. Gershunov A, Barnett P T. 1997. ENSO influence on intraseasonal extreme rainfall and temperature frequencies in the contiguous United States: Observations and model results Journal of Climate 11 : 15751586. Goto Maede Y, Shin DW, OBrien JJ. 2008 Freeze probability of Florida in a regional climate model and climate indices. Geophysical Research Letters 35 : L11703, D oi:10.1029/2008GL033720.

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61 BIOGRAPHICAL SKETCH David James Keellings was born in 1986 in Glasgow, Scotland. He grew up in the small town of Kirkintilloch and attended primary and secondary school in the adjacent village of Lenzie. In 2001 he moved to Orlando, Florida where he finished his high school education at Trinity Preparatory School of Winter Park He received a Bachelor of Science d egree with Honors in environmental s tudies from the Universi ty of Central Florida in 2007. Throughout this time he has worked as an environmental consultant on numerous development projects around Centr al Florida. David hopes to remain in the Geography Department at t he University of Florida to pursue a doctorate d egree under the advisement of Dr. Peter Waylen.